Hamiltonian structure of isomonodromic deformation dynamics in linear systems of PDE's

Abstract

The Hamiltonian approach to isomonodromic deformation systems for generic rational covariant derivative operators on the Riemann sphere, having any matrix dimension $r$ and any number of isolated singularities of arbitrary Poincar\'e rank, is derived using the split classical rational $R$-matrix Poisson bracket structure on the dual space $L^*{\frak gl}(r)$ of the loop algebra $L{\frak gl}(r)$. Nonautonomous isomonodromic counterparts of isospectral systems are obtained by identifying the deformation parameters as Casimir elements on the phase space. These are shown to coincide with the higher Birkhoff invariants determining the local asymptotics near to irregular singular points, together with the pole loci. They appear as the negative power coefficients in the principal part of the Laurent expansion of the fundamental meromorphic differential on the associated spectral curve, while the corresponding dual spectral invariant Hamiltonians appear as the "mirror image" positive power terms in the analytic part. Infinitesimal isomonodromic deformations are generated by the sum of the Hamiltonian vector field and an explicit derivative vector field that is transversal to the symplectic foliation.